Creating Symmetry The Artful Mathematics of Wallpaper Patterns

"Frank's rich artistry vibrantly enlivens the mathematics of symmetry. What a treat for the eye and the mind!"--Michael Starbird, The University of Texas at Austin

"The imagery in this book is attractive and engaging, and illustrates Farris's excellent command of the mathematical techniques and his strong sense of visual design. General readers will appreciate the beauty of the images and will find incentive to learn how mathematics can be used in creative ways to produce art."--George W. Hart, coauthor of Zome Geometry: Hands-on Learning with Zome Models

"This unique book takes an entirely new approach to creating images with symmetry. The pictures are compelling and Farris presents the material in an inviting manner. He leads readers into interesting areas of mathematics not usually encountered in undergraduate courses, and rarely, if ever, encountered as a way to study symmetry."--Doris Schattschneider, author of M. C. Escher: Visions of Symmetry

"In Creating Symmetry, Farris explores the concept of symmetry and its application to creating artistic patterns in two dimensions. The result is a set of algorithmic tools for transforming ordinary photographs into rosettes and wavelike murals decorated with colorful swirls and gradients. It is difficult to imagine a more engaging focus for teaching the mathematics of symmetry."--Kenneth Libbrecht, author of The Secret Life of a Snowflake: An Up-Close Look at the Art and Science of Snowflakes

"Creating Symmetry is a stunning fusion of mathematics and art, applying the mathematics of symmetry to create beautiful patterns. But the beauty runs far deeper: the mathematical insights involved are supremely beautiful in their own right. If you want to know why there are exactly 17 basic types of wallpaper, what their structure is, and which other mathematical ideas are related, or if you just want to see some amazing pictures, look no further."--Ian Stewart, author of Professor Stewart's Casebook of Mathematical Mysteries

"Creating Symmetry is a remarkable, one-of-a-kind book, with unique and beautiful pictures. While plane symmetry groups have been the subject of other books, Farris's approach is fresh and accessible."--John Stillwell, author of Roads to Infinity: The Mathematics of Truth and Proof

"Farris has written an amazing book. His vision is expansive, his enthusiasm is contagious, and the illustrations are intriguing and beautiful. Farris enables readers to gain a deep appreciation and understanding of the mathematics behind symmetry and his novel approach to creating symmetrical patterns. No other book comes close."--Thomas Q. Sibley, author of Foundations of Mathematics

"This book introduces readers to the fascinating interplay of geometry, complex function theory, abstract algebra, complex domain coloring, Fourier series, and aesthetics in producing really beautiful images. Farris shows how structured forms of symmetry can be constructed in a disciplined way from first mathematical principles, and how artistically pleasing images can communicate sophisticated but understandable mathematics."--Paul Zorn, author of Understanding Real Analysis

Farris targets three types of readers--the working mathematician, theadvanced undergraduate, and the brave mathematical adventurer--Hoping to hook them with the abstract beauty ofmathematically generated wallpaper patterns without burdening them with extensive mathematics beyond calculus. Farris promises that ifthe reader is willing to learn they can have a fine experience working through his story of symmetry, and while the mathematicalcontent of the book is rather sophisticated, everything can be approached from an elementary point of view. Working with 17different wallpaper types, he advances into more complex math--functional analysis, group theory, and partial differentialequations--all leading to the reader creating art of their own. A sampling of the twenty-seven chapters follow: complex numbers androtations; symmetry of the mystery curve; mathematical structures and symmetry; fourier series; rosettes as plane functions; plane wavepackets for 3-fold symmetry; waves, mirrors, and 3-fold symmetry; wallpaper groups and 3–fold symmetry; wallpaper with a squarelattice; color-turning wallpaper functions; local symmetry in wallpaper and rings of integers; hyperbolic wallpaper; morphing friezes and mathematical art; and an epilogue.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

Farris targets three types of readers--the working mathematician, theadvanced undergraduate, and the brave mathematical adventurer--Hoping to hook them with the abstract beauty ofmathematically generated wallpaper patterns without burdening them with extensive mathematics beyond calculus. Farris promises that ifthe reader is willing to learn they can have a fine experience working through his story of symmetry, and while the mathematicalcontent of the book is rather sophisticated, everything can be approached from an elementary point of view. Working with 17different wallpaper types, he advances into more complex math--functional analysis, group theory, and partial differentialequations--all leading to the reader creating art of their own. A sampling of the twenty-seven chapters follow: complex numbers androtations; symmetry of the mystery curve; mathematical structures and symmetry; fourier series; rosettes as plane functions; plane wavepackets for 3-fold symmetry; waves, mirrors, and 3-fold symmetry; wallpaper groups and 3–fold symmetry; wallpaper with a squarelattice; color-turning wallpaper functions; local symmetry in wallpaper and rings of integers; hyperbolic wallpaper; morphing friezes and mathematical art; and an epilogue.Annotation ©2015 Ringgold, Inc., Portland, OR (protoview.com)

"[A] beautiful book. . . . [Creating Symmetry] is a thoughtful, innovative and interesting piece of work, discussing material that the author is obviously very enthusiastic about; such enthusiasm is, as is often the case, contagious."--Mark Hunacek, MAA Reviews

"This is a marvelous book that brings groups, and along the way many other mathematical concepts, to the reader in an unconventional way."--Adhemar Bultheel, European Mathematical Society Bulletin

This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks--a sort of potato-stamp method--Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.

Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own.

Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.

Frank A. Farris teaches mathematics at Santa Clara University. He is a former editor of Mathematics Magazine, a publication of the Mathematical Association of America. He lives in San Jose, California.

Preface vii

1 Going in Circles 1

2 Complex Numbers and Rotations 5

3 Symmetry of the Mystery Curve 11

4 Mathematical Structures and Symmetry: Groups, Vector Spaces, and More 17

5 Fourier Series: Superpositions of Waves 24

6 Beyond Curves: Plane Functions 34

7 Rosettes as Plane Functions 40

8 Frieze Functions (from Rosettes!) 50

9 Making Waves 60

10 PlaneWave Packets for 3-Fold Symmetry 66

11 Waves, Mirrors, and 3-Fold Symmetry 74

12 Wallpaper Groups and 3-Fold Symmetry 81

13 ForbiddenWallpaper Symmetry: 5-Fold Rotation 88

14 Beyond 3-Fold Symmetry: Lattices, Dual Lattices, andWaves 93

15 Wallpaper with a Square Lattice 97

16 Wallpaper with a Rhombic Lattice 104

17 Wallpaper with a Generic Lattice 109

18 Wallpaper with a Rectangular Lattice 112

19 Color-ReversingWallpaper Functions 120

20 Color-Turning Wallpaper Functions 131

21 The Point Group and Counting the 17 141

22 Local Symmetry in Wallpaper and Rings of Integers 157

23 More about Friezes 168

24 Polyhedral Symmetry (in the Plane?) 172

25 HyperbolicWallpaper 189

26 Morphing Friezes and Mathematical Art 200

27 Epilog 206

A Cell Diagrams for the 17 Wallpaper Groups 209

B Recipes forWallpaper Functions 211

C The 46 Color-ReversingWallpaper Types 215

Bibliography 227

Index 229